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A086990
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Expansion of (1+4x-sqrt(1+4x^2))/(4+6x) in powers of x.
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4
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0, 1, -2, 3, -4, 6, -10, 15, -20, 30, -52, 78, -96, 144, -282, 423, -420, 630, -1660, 2490, -1304, 1956, -11332, 16998, 3896, -5844, -95240, 142860, 157160, -235740, -983610, 1475415, 2634300, -3951450, -11751660, 17627490, 38381160, -57571740
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OFFSET
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0,3
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COMMENTS
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Series reversion of Sum_{k>=0} a(k)x^k is x(Sum_{k>=0} A007051(k)x^k).
G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = x - (4x+1)*A(x) + (3x+2)*A(x)^2.
If A(x)=g.f., then y=x/A(x)-2x satisfies x^2 = y^2 - y.
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LINKS
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FORMULA
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G.f.: (1+4x-sqrt(1+4x^2))/(4+6x).
G.f.: (x-x^2*c(-x^2))/(1+x-x^2*c(-x^2)), c(x) the g.f. of A000108. - Paul Barry, Jun 17 2005
a(n) is the upper left term of (-1)*M^n, where M = an infinite square production matrix as follows:
-1, -1, 0, 0, 0, 0, ...
-1, 1, -1, 0, 0, 0, ...
-1, 1, 1, -1, 0, 0, ...
-1, 1, 1, 1, -1, 0, ...
-1, 1, 1, 1, 1, -1, ...
... (End)
D-finite with recurrence 2*n*a(n) +3*n*a(n-1) +8*(n-3)*a(n-2) +12*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
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EXAMPLE
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MATHEMATICA
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CoefficientList[Series[(1 + 4 x - Sqrt[1 + 4 x^2])/(4 + 6 x), {x, 0, 50}], x] (* Harvey P. Dale, Mar 24 2011 *)
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PROG
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(PARI) a(n)=polcoeff((1+4*x-sqrt(1+4*x^2+x*O(x^n)))/(4+6*x), n)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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